Add GitHub Actions workflow to trigger docs-staging nextgen update

[qiancai]

First-time contributors' checklist

• I've signed Contributor License Agreement that's required for repo owners to accept my contribution.

What is changed, added, or deleted? (Required)

Which TiDB Operator version(s) do your changes apply to? (Required)

• main (the latest development version for v2.x)
• v2.0 (TiDB Operator 2.0 versions)
• release-1.x (the latest development version for v1.x)
• v1.6 (TiDB Operator 1.6 versions)
• v1.5 (TiDB Operator 1.5 versions)
• v1.4 (TiDB Operator 1.4 versions)
• v1.3 (TiDB Operator 1.3 versions)

What is the related PR or file link(s)?

• This PR is translated from:
• Other reference link(s): Add GitHub Actions workflow to trigger docs-staging nextgen update docs#22342

[ti-chi-bot]

[LGTM Timeline notifier]

Timeline:

• 2026-01-19 03:33:16.006387912 +0000 UTC m=+378423.620344758: ☑️ agreed by lilin90.
• 2026-01-19 03:35:25.971421481 +0000 UTC m=+378553.585378327: ☑️ agreed by Oreoxmt.

[qiancai]

/approve

[ti-chi-bot]

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